As the Killer:

Since the investigator cannot change their identity, early on, try to eliminate possible suspects for the investigator by noting who's been exonerated, killed, or arrested so far. When killing, prioritize exonerated citizens.

If Citizen's arrest is on, prioritize not being caught, as the investigator can potentially be holding any adjacent card to arrest you. Try shifting rows and columns to disguise your position. Without Citizen's arrest, you can get away with killing more people before having to disguise.

As the Investigator:

Don't give yourself away too early, but don't be afraid to guess on arrests either. Note that besides anyone who's dead/exonerated, you can also eliminate anyone in your hand. Try to align rows or columns of deceased people to narrow the field and exonerate more at a time.